Question

find \\(f+g\\), \\(f-g\\), \\(fg\\), \\(\frac{f}{g}\\). determine the domain for each function.

\\f(x) = \frac{5x}{x-4}, g(x) = \frac{4}{x+5}\\

what is the domain of \\(f-g\\)?

a. the domain of \\(f-g\\) is \\((-\infty, -5) \cup (-5, 4) \cup (4, \infty)\\) (type your answer in interval notation)
b. the domain of \\(f-g\\) is \\(\\{\quad\\}\\) (use a comma to separate answers as needed.)
c. the domain of \\(f-g\\) is \\(\varnothing\\).

\\((fg)(x) = \quad\\) (simplify your answer.)

what is the domain of \\(fg\\)?

a. the domain of \\(fg\\) is \\(\quad\\) (type your answer in interval notation.)
b. the domain of \\(fg\\) is \\(\\{\quad\\}\\) (use a comma to separate answers as needed.)

Explanation:

Find the domains of the individual functions

Using the Domain of Rational Functions knowledge point

$$
LATEXBLOCK0
$$

Determine the domain of the difference function

Using the Function Operations and Interval Notation knowledge points

$$
LATEXBLOCK1
$$

Find the product function

Using the Function Operations knowledge point

$$ (fg)(x) = f(x) \cdot g(x) = \frac{5x}{x-4} \cdot \frac{4}{x+5} = \frac{20x}{(x-4)(x+5)} $$

Determine the domain of the product function

Using the Function Operations and Interval Notation knowledge points

$$
LATEXBLOCK2
$$

Answer:

Question 1

• A. The domain of \(f - g\) is \((-\infty, -5) \cup (-5, 4) \cup (4, \infty)\) (Correct answer)
• B. The domain of \(f - g\) is { }
• C. The domain of \(f - g\) is \(\varnothing\)

Question 2

\((fg)(x) =\) <blank>\(\frac{20x}{(x-4)(x+5)}\)</blank>

Question 3

• A. The domain of \(fg\) is \((-\infty, -5) \cup (-5, 4) \cup (4, \infty)\) (Correct answer)
• B. The domain of \(fg\) is { }